Wigner distribution ($W[u,v](\mathbf{z},\mathbf{p})$)

The Wigner distribution of two fields is the Fourier transform of a field's two-point product in its separation variable:Its inverse reconstructs $u(\mathbf{z})\overline{v({\mathbf{z}}^{\prime })}$. For vector fields, specify whether the product is an outer product or a scalar contraction; the latter can encode an average over receiver channels.

For receiver-dependent Green functions $g_{\mathbf{\zeta }}(\mathbf{z})$ and aperture weight $a(\mathbf{\zeta })$, define ${\mathcal{W}}_A(\mathbf{Z},\mathbf{p})=\int a(\mathbf{\zeta })W[g_{\mathbf{\zeta }},g_{\mathbf{\zeta }}](\mathbf{Z},\mathbf{p})d\mathbf{\zeta }$. Its inverse separation Fourier transform is the kernel of the time reversal operator on the source plane.